INDEPENDENCE and WRONSKIANS

Consider equations of the form $L[y]=\frac{d^2y}{dt^2}+ p(t)\frac{dy}{dt}+ q(t)y = 0,$
for $t, t_0\in I$, with $p, q \in C(I)$, and y(t₀)=y₀, y'(t₀)=y'₀.

Linear Independence:

functions f and g are
linearly dependent if there exist constants k₁ and k₂, not both zero, so that k₁f(t) + k₂g(t) = 0 for all $t \in I$;
f and g are linearly independent if not dependent.
If f(t) = cg(t) for some c, f and g are dependent.

Wronskians and Independence

• If $W(f,g)(t_0)\neq 0$, one $t_0\in I$, f and g independent.
• If f and g dependent, then W(f,g)(t)=0 for all $t \in I$.
• Abel's Theorem: if L[y₁]=0 and L[y₂]=0, then
  
  $$W(y_1,y_2)(t) = Ce^{-\displaystyle\int p(t) dt},$$
  
  
  where C depends on y₁ and y₂, but not t;
  either W(t) = 0 for all $t \in I$ or $W(t) \neq 0$ for all $t \in I$.
• If L[y₁]=0 and L[y₂]=0, then y₁ and y₂ are linearly dependent if and only if W(y₁,y₂)(t) = 0 for all $t \in I$.

Summary: Equivalent Conditions

• Functions y₁ and y₂ are fundamental solutions on I;
• Functions y₁ and y₂ are linearly independent on I;
• $W(y_1,y_2)(t_0) \neq 0$ for some $t_0\in I$;
• $W(y_1,y_2)(t) \neq 0$ for all $t \in I$.

The Kernel of L:

solutions to L[y]=0 form a
two-dimensional vector space: the kernel of L.